Linear response function of a many-fermion system

Abstract

The properties of a many-body system are studied by means of a “linear response function” which depends on frequency and wave number. It is shown how the expectation value of two-body operators, the rate of transitions induced by one-body operators or weak external fields acting on the system, and information about the energy spectrum can be found from this function. As a result, this formulation suggests many relationships between apparently dissimilar quantities, and suggests indirect ways of experimentally measuring internal properties of a system. In the present formalism, all of the difficulties inherent in many body calculations are relegated to the determination of the response function. However, there exist sum rules and a priori conditions on this function which can be used to estimate the validity of approximate calculations and to indicate what must be done to improve the results. The response function is calculated for a Fermi gas by using a combined diagrammatic perturbation theory and Green's function technique. Expressing the response function in terms of the irreducible particle-hole propagator, it is shown that a many fermion system with repulsive interactions between particles can be expected to respond strongly at certain frequencies at least for disturbances of very small wave number. This enhanced response can be associated with the excitation of a plasmon or second sound. It is also shown that the approximations usually made in treating the electron gas lead to a response function which violates a priori conditions. The corrections needed to remedy this defect are indicated.